Discussion
Let \(|\vec{a}| = 6\) and \(|\vec{b}| = 10\). If \(\vec{a}\) and \(\vec{b}\) make angles 25° and 85°, respectively, with the x-axis, then the value of \(|\vec{a} + \vec{b}|\) is equal to
Answer: A
\(|\vec{a} + \vec{b}|\) = 14.
The angle between the vectors is 85° − 25° = 60°.
\[ \begin{aligned} |\vec{a} + \vec{b}|^{2} &= 36 + 100 + 2(6)(10)\cos 60^{\circ} \\ &= 136 + 60 = 196 \\ |\vec{a} + \vec{b}| &= 14 \end{aligned} \]
Using the angle 110° (the sum of the two angles) or forgetting the cross term would give values not offered. Subtracting instead of adding the cross term gives \(\sqrt{76}\), the magnitude of \(\vec{a} - \vec{b}\).
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